
As noted by [1]}, [2]} and  [3]}, the above problem is a stochastic 3-level composition optimization problem with
\(f_3 (x) (x , \mathbb {E}[U(x,\xi )]),~ f_2(y_3, y_4) \left(y_4, \mathbb {E}\left[ \max \left\lbrace  0 , U(y_3,\xi ) - y_4 \right\rbrace ^2 \right] \right), ~f_1(y_1,y_2) y_1 + \sqrt{y_2+\delta }.\) 
